- Email: [email protected]
Analysis of electrical relaxation in ionically conducting glasses and melts
JOURNAI, OF
ELSEVIER
Journal of Non-Crystalline Solids 203 (1996) 359-363
Analysis of electrical relaxation in ionically conducting glasses and melts Cornelius T. M o y n i h a n * Materials Science and Engineering Department, RensselaerPolytechnic Institute, Troy NY 12180-3590, USA
Abstract The Havriliak-Negami (HN) relaxation function was used to analyze electric field relaxation data for a model Li20-A1203-2SiO 2 glass. The HN function gave an excellent description of the frequency dependence of the imaginary part of the electric modulus M" in the intermediate frequency range (10 -~- < w ( r ) < 102). However, at low frequencies the HN function predicted unphysical behavior, i.e., it failed to predict the levelling off of the real parts of the complex conductivity, o-', and of the complex permittivity, e', at static low frequency values. It appears that, for fits using a minimal number of adjustable parameters, the distribution of relaxation times associated with the KWW or stretched exponential relaxation function, q50)= exp(-t/'rKWW )13, continues to give the best description of electrical relaxation in materials containing large concentrations of mobile ions.
1. Introduction Electrical relaxation in liquids, glasses and crystals containing high concentrations of mobile ions, as manifested in, for example, the frequency dependence of the real parts of the complex conductivity, or', and complex permittivity, e ' , is generally attributed to motions of the mobile ions whose long range transport is responsible for the dc conductivity, o-. In a recent paper [1] we compared various empirical analyses of this electrical relaxation using a model set of data obtained at 24.0°C for a lithium aluminosilicate ( L A S ) glass of composition L i z O A I ~ O 3 - S i O 2 [2,3]. The relaxation of the electric
field E under the constraint of constant displacement vector was assumed to be of the form ~c
E( t) = E(O) d)( t) =- E(O) f ° g ( ~ - ) e x p ( - t / r )
dT,
(l) where E(0) is the initial electric field imposed at time t = 0 and ¢ ( t ) the electric field relaxation function. ¢b(t) was in turn assumed to be expressible in terms of a distribution of electric field relaxation times, T, where g(~-) is the normalized probability density function for T. In the frequency domain Eq. (1) takes the form 3C
M* = M ~ f ° g(T)[ioor/(1 + i w ' c ) ] d T , * Tel.: +1-518 276 6125; fax: +1-518 276 8554; e-mail: [email protected].
(2)
where o~ is angular frequency and M * is usually termed the electric modulus. M * is in turn related to
0022-3093/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PI1 S0022-3093(96)00501-7
360
C.T. Moynihan / Journal of Non-Crystalline Solids 203 (1996) 359-363
the complex permittivity, e *, conductivity, o- *, and resistivity, p * , by M* = l / e *
= iwe0/~*
= ioge 0 p * ,
(3)
where e o is the permittivity of free space and M~=
lim M* = 1 , / lim e* = l / e ~ .
(4)
A minimal description in terms of Eq. (2) of the electric field relaxation in an ionically conducting material requires at least three adjustable parameters specifying (i) the 'strength' of the relaxation M~, (ii) the timescale of the relaxation in terms of some reference relaxation time or in terms of the mean relaxation time ( r ) , related to the dc conductivity oby ( r ) = ~o/M~o-
(5)
and (iii) the weighting of the other relaxation times r relative to the reference relaxation time, i.e., the distribution of relaxation times. In our previous paper [1] we compared fits to the LAS glass data obtained with three different empirical expressions specifying the effective distribution of electric field relaxation times. The first two of these were three parameter fits using the distributions of relaxation times implicit in the K W W or stretched exponential relaxation function, 4)(t) = exp( - t / r K w w) ~
(6)
with 0 < / 3 < 1, and in the Cole-Davidson (CD) expression for M *. The third was a four parameter fit using the distribution of relaxation times implicit in the Jonscher expressions for the frequency dependence of the real parts of the complex conductivity and permittivity: or' = o ' + A w "
(7a)
e ' = e~ + [ a / g o c o t ( n r r / 2 ) ] O.)n-1
(7b)
where A is a constant and 0 < n < 1. The quality of each fit was judged by how well it described the data quantitatively in the reduced frequency range 10 -2 < w ( r ) < 10 2, where most of the relaxation takes place, as well as by whether or not the fit qualitatively replicated the frequency dependence of M *, e *, o-* and p * over a more extended frequency range. It was found that the three parameter K W W fit gave the best quantitative agreement in the 10 -2
~o(r) ~ 10 2 range, as well as qualitative agreement over the more extended frequency range. The four parameter Jonscher fit, on the other hand, was found to be in qualitative disagreement with the experimental e * and p * data at low frequencies. In a paper published several years ago Cole and co-workers [4] noted that good fits to sodium silicate and lithium fluoroborate glass electrical relaxation data in the M * form could be achieved using the Havriliak-Negami relaxation function. In the present paper we evaluate the use of this relaxation function in the analysis of the LAS glass data considered in our earlier paper.
2. Fits using the HavrUiak-Negami expression The empirical Havriliak-Negami (HN) relaxation function was originally devised to give an accurate portrayal of the shapes of e" versus e' complex plane plots for polymer dielectric relaxation data [5]. It may be considered a hybrid of the Cole-Cole and Cole-Davidson relaxation functions, which predict e" versus ~' arcs which are, respectively, symmetric and highly asymmetric. For the electric modulus M * the HN function takes the form M* = M ' + i M " = M ~ [ I - r
y/2
Xcos(3,0) + ir-V/2 sin(3,0)] ,
(8) where r = [1 + (¢.0THN) 1
~sin( ffTr/2)] 2
+ [ ( ~ ' r H y ) l - a c o s ( c e v ' r / 2 ) ] 2,
0 = arctan
(9)
( WrHN)I ~ c o s ( a r t ~ 2 ) 1 q- ( O)"/'HN)1
a sin(c~rr/2)
(m)
In the above expressions THN is a reference relaxation time, while the parameters a (0 < c~ < 1) and 3' (0 < 3' < 1) effectively characterize the distribution of relaxation times. The HN function thus involves the use of four adjustable parameters: Ms, THN, a and 3'-
c. T. Moynihan /Journal of Non-Co,stalline Solids 203 (1996) 359-363
The solid line in Fig. 1 is the M" versus f curve calculated using these parameters. Also shown in Fig. 1 is the M" versus f curve (the dashed line) obtained in Ref. [1] from the best fit parameters using the K W W relaxation function of Eq. (6):
Shown in Fig. 1 is the experimental plot of the imaginary part of the electric modulus, M", versus frequency, f, for the LAS glass at 24.0°C. To fit this data to the HN function, the value of M~ was set equal to the experimental value obtained by extrapolation of the complex plane plot of M" versus M' to high frequency (see fig. 6 in Ref. [1]). The THN, a and y parameters were then determined by requiring the best match between the calculated and experimental values of three (f, M") points in the frequency range 10 -2 G W(7") G 10 2, in particular, the (f, M") point at the maximum and the two (f, M") points at the half-height positions on the M" versus f curve. This yielded the following parameters for the HN fit:
=0.118, THN = 2 . 1 0 X
M~ = 0.110,
(12b) /3 = 0.47.
(llc)
y = 0.33.
(lld)
(12c)
Plainly the four parameter HN fit gives a much better description of the M" versus f data in Fig. 1 than does the three parameter K W W fit. However, as emphasized in Ref. [1], it is important to check the quality of the fits when plotting the electrical relaxation data in the other e *, ~r * and p * formats. To this end a plot of the real part, o-', of the complex conductivity versus frequency is shown in Fig. 2, along with curves calculated via Eq. (3) from the HN and K W W fit parameters. Both the HN and K W W
(lib)
a=0.10,
(12a)
~'KWW= / 3 e o / M ~ ° ' f f ( 1 / / 3 ) = 3.92 X 10 .4 s,
(lla) 10 3 S,
361
0.03
,,
Li20 - AI203 - 2SiO 2
0.02
M" 0.015
0.01
0.005
log f(Hz) Fig. 1. Imaginary part of the electric modulus M" versus frequency for LAS glass at 24.0°C. Points are experimental data. Lines are calculated from the HN and KWW fit parameters in Eqs. (11) and (12).
C.T. Moynihan / Journal of Non-Crystalline Solids 203 (1996) 359-363
362
-6
Li20 - AI203 - 2SiO 2 -6.5
o /
24"0°C
o°///'"
.--~__ : N f i t
-7-
/Z"/'"'"
fit
-7.5-
? -8-
w=
la0
1
, / / ' " ' " "
-8.5-
-9-
-9.s-1
....
; ....
i
....
t
....
~ ....
~ ....
+ ....
~ ....
log y(Hz) Fig. 2. Real part o f the c o m p l e x conductivity o-' versus f r e q u e n c y for L A S glass at 24.0°C. Points are experimental data. Lines are calculated f r o m the H N a n d K W W fit parameters in Eqs. (l 1) a n d (12).
fits predict qualitatively the observed fractional exponential dependence of ~r' on frequency in the high frequency range (oJ(~') >> 1): lim (dlog ~r'/dlog w) = 1 - (1 - c~)y for HN, (13a) lim (dlog o-'/dlog w) = 1 - 13 for KWW.
o) .-.-)oc
(13b) However, at low frequencies the HN fit, unlike the KWW fit, fails to predict the levelling off of the o-' versus f plot at the dc conductivity o'. Rather, at low frequencies the HN fit leads to a continual decrease of o-' with decreasing frequency with limiting slope: lim (dlog o " / d l o g w) = a.
(14)
co--~ 0
In similar fashion, the HN fit predicts at low frequencies a continual rise with decreasing frequency of the real part of the complex permittivity e', lim (dlog e ' / d l o g w) = c~- 1, ~o---*0
(15)
rather than a levelling off of e' at a static low frequency value, as predicted by the KWW fit and observed experimentally [1]. Finally, the complex plane resistivity plot of p" versus p' predicted from the HN fit does not converge on the dc resistivity p ( = 1/o-) at low frequencies, as observed experimentally and predicted by the KWW fit, but rather diverges with decreasing frequency. Because of the failure of the HN relaxation function to account correctly in a qualitative fashion for the observed low frequency electrical relaxation behavior, it must be judged pathological and unsuited for the description of electric field relaxation in ionically conducting materials.
3. Conclusions In agreement with the conclusions reached in our earlier paper [1], it appears that the KWW distribution of relaxation times remains the most accurate description of electrical relaxation in ionically con-
C.T. Moynihan / Journal of Non-Crystalline Solids 203 (1996) 359-363
ducting melts, glasses and crystals, if one wishes to describe the data in the frequency region where most of the relaxation is taking place in terms of a minimal number of parameters. We should emphasize, as before, that the value of the KWW/3 parameter is a description and not an explanation of the electrical relaxation process. The utility of the K W W / 3 parameter is that it can be used as a simple index for charting correlations and systematizing trends in the shapes of electrical relaxation curves as a function of variables such as temperature and glass or melt composition.
363
References [1] C.T. Moynihan, J. Non-Cryst. Solids 172-174 (1994) 1395. [2] P.B. Macedo, C.T. Moynihan and R. Bose, Phys. Chem. Glasses 13 (1972) 171. [3] C.T. Moynihan, L.P. Boesch and N.L. Laberge, Phys. Chem. Glasses 14 (1973) 122. [4] A. Bums, G.D. Chryssikos, E. Tombari, R.H. Cole and W.M. Risen Jr., Phys. Chem. Glasses 30 (1989) 264. [5] S. Havriliak and S. Negami, J. Polym. Sci. C 14 (1966) 99.
ELSEVIER
Journal of Non-Crystalline Solids 203 (1996) 359-363
Analysis of electrical relaxation in ionically conducting glasses and melts Cornelius T. M o y n i h a n * Materials Science and Engineering Department, RensselaerPolytechnic Institute, Troy NY 12180-3590, USA
Abstract The Havriliak-Negami (HN) relaxation function was used to analyze electric field relaxation data for a model Li20-A1203-2SiO 2 glass. The HN function gave an excellent description of the frequency dependence of the imaginary part of the electric modulus M" in the intermediate frequency range (10 -~- < w ( r ) < 102). However, at low frequencies the HN function predicted unphysical behavior, i.e., it failed to predict the levelling off of the real parts of the complex conductivity, o-', and of the complex permittivity, e', at static low frequency values. It appears that, for fits using a minimal number of adjustable parameters, the distribution of relaxation times associated with the KWW or stretched exponential relaxation function, q50)= exp(-t/'rKWW )13, continues to give the best description of electrical relaxation in materials containing large concentrations of mobile ions.
1. Introduction Electrical relaxation in liquids, glasses and crystals containing high concentrations of mobile ions, as manifested in, for example, the frequency dependence of the real parts of the complex conductivity, or', and complex permittivity, e ' , is generally attributed to motions of the mobile ions whose long range transport is responsible for the dc conductivity, o-. In a recent paper [1] we compared various empirical analyses of this electrical relaxation using a model set of data obtained at 24.0°C for a lithium aluminosilicate ( L A S ) glass of composition L i z O A I ~ O 3 - S i O 2 [2,3]. The relaxation of the electric
field E under the constraint of constant displacement vector was assumed to be of the form ~c
E( t) = E(O) d)( t) =- E(O) f ° g ( ~ - ) e x p ( - t / r )
dT,
(l) where E(0) is the initial electric field imposed at time t = 0 and ¢ ( t ) the electric field relaxation function. ¢b(t) was in turn assumed to be expressible in terms of a distribution of electric field relaxation times, T, where g(~-) is the normalized probability density function for T. In the frequency domain Eq. (1) takes the form 3C
M* = M ~ f ° g(T)[ioor/(1 + i w ' c ) ] d T , * Tel.: +1-518 276 6125; fax: +1-518 276 8554; e-mail: [email protected].
(2)
where o~ is angular frequency and M * is usually termed the electric modulus. M * is in turn related to
0022-3093/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PI1 S0022-3093(96)00501-7
360
C.T. Moynihan / Journal of Non-Crystalline Solids 203 (1996) 359-363
the complex permittivity, e *, conductivity, o- *, and resistivity, p * , by M* = l / e *
= iwe0/~*
= ioge 0 p * ,
(3)
where e o is the permittivity of free space and M~=
lim M* = 1 , / lim e* = l / e ~ .
(4)
A minimal description in terms of Eq. (2) of the electric field relaxation in an ionically conducting material requires at least three adjustable parameters specifying (i) the 'strength' of the relaxation M~, (ii) the timescale of the relaxation in terms of some reference relaxation time or in terms of the mean relaxation time ( r ) , related to the dc conductivity oby ( r ) = ~o/M~o-
(5)
and (iii) the weighting of the other relaxation times r relative to the reference relaxation time, i.e., the distribution of relaxation times. In our previous paper [1] we compared fits to the LAS glass data obtained with three different empirical expressions specifying the effective distribution of electric field relaxation times. The first two of these were three parameter fits using the distributions of relaxation times implicit in the K W W or stretched exponential relaxation function, 4)(t) = exp( - t / r K w w) ~
(6)
with 0 < / 3 < 1, and in the Cole-Davidson (CD) expression for M *. The third was a four parameter fit using the distribution of relaxation times implicit in the Jonscher expressions for the frequency dependence of the real parts of the complex conductivity and permittivity: or' = o ' + A w "
(7a)
e ' = e~ + [ a / g o c o t ( n r r / 2 ) ] O.)n-1
(7b)
where A is a constant and 0 < n < 1. The quality of each fit was judged by how well it described the data quantitatively in the reduced frequency range 10 -2 < w ( r ) < 10 2, where most of the relaxation takes place, as well as by whether or not the fit qualitatively replicated the frequency dependence of M *, e *, o-* and p * over a more extended frequency range. It was found that the three parameter K W W fit gave the best quantitative agreement in the 10 -2
~o(r) ~ 10 2 range, as well as qualitative agreement over the more extended frequency range. The four parameter Jonscher fit, on the other hand, was found to be in qualitative disagreement with the experimental e * and p * data at low frequencies. In a paper published several years ago Cole and co-workers [4] noted that good fits to sodium silicate and lithium fluoroborate glass electrical relaxation data in the M * form could be achieved using the Havriliak-Negami relaxation function. In the present paper we evaluate the use of this relaxation function in the analysis of the LAS glass data considered in our earlier paper.
2. Fits using the HavrUiak-Negami expression The empirical Havriliak-Negami (HN) relaxation function was originally devised to give an accurate portrayal of the shapes of e" versus e' complex plane plots for polymer dielectric relaxation data [5]. It may be considered a hybrid of the Cole-Cole and Cole-Davidson relaxation functions, which predict e" versus ~' arcs which are, respectively, symmetric and highly asymmetric. For the electric modulus M * the HN function takes the form M* = M ' + i M " = M ~ [ I - r
y/2
Xcos(3,0) + ir-V/2 sin(3,0)] ,
(8) where r = [1 + (¢.0THN) 1
~sin( ffTr/2)] 2
+ [ ( ~ ' r H y ) l - a c o s ( c e v ' r / 2 ) ] 2,
0 = arctan
(9)
( WrHN)I ~ c o s ( a r t ~ 2 ) 1 q- ( O)"/'HN)1
a sin(c~rr/2)
(m)
In the above expressions THN is a reference relaxation time, while the parameters a (0 < c~ < 1) and 3' (0 < 3' < 1) effectively characterize the distribution of relaxation times. The HN function thus involves the use of four adjustable parameters: Ms, THN, a and 3'-
c. T. Moynihan /Journal of Non-Co,stalline Solids 203 (1996) 359-363
The solid line in Fig. 1 is the M" versus f curve calculated using these parameters. Also shown in Fig. 1 is the M" versus f curve (the dashed line) obtained in Ref. [1] from the best fit parameters using the K W W relaxation function of Eq. (6):
Shown in Fig. 1 is the experimental plot of the imaginary part of the electric modulus, M", versus frequency, f, for the LAS glass at 24.0°C. To fit this data to the HN function, the value of M~ was set equal to the experimental value obtained by extrapolation of the complex plane plot of M" versus M' to high frequency (see fig. 6 in Ref. [1]). The THN, a and y parameters were then determined by requiring the best match between the calculated and experimental values of three (f, M") points in the frequency range 10 -2 G W(7") G 10 2, in particular, the (f, M") point at the maximum and the two (f, M") points at the half-height positions on the M" versus f curve. This yielded the following parameters for the HN fit:
=0.118, THN = 2 . 1 0 X
M~ = 0.110,
(12b) /3 = 0.47.
(llc)
y = 0.33.
(lld)
(12c)
Plainly the four parameter HN fit gives a much better description of the M" versus f data in Fig. 1 than does the three parameter K W W fit. However, as emphasized in Ref. [1], it is important to check the quality of the fits when plotting the electrical relaxation data in the other e *, ~r * and p * formats. To this end a plot of the real part, o-', of the complex conductivity versus frequency is shown in Fig. 2, along with curves calculated via Eq. (3) from the HN and K W W fit parameters. Both the HN and K W W
(lib)
a=0.10,
(12a)
~'KWW= / 3 e o / M ~ ° ' f f ( 1 / / 3 ) = 3.92 X 10 .4 s,
(lla) 10 3 S,
361
0.03
,,
Li20 - AI203 - 2SiO 2
0.02
M" 0.015
0.01
0.005
log f(Hz) Fig. 1. Imaginary part of the electric modulus M" versus frequency for LAS glass at 24.0°C. Points are experimental data. Lines are calculated from the HN and KWW fit parameters in Eqs. (11) and (12).
C.T. Moynihan / Journal of Non-Crystalline Solids 203 (1996) 359-363
362
-6
Li20 - AI203 - 2SiO 2 -6.5
o /
24"0°C
o°///'"
.--~__ : N f i t
-7-
/Z"/'"'"
fit
-7.5-
? -8-
w
la0
1
, / / ' " ' " "
-8.5-
-9-
-9.s-1
....
; ....
i
....
t
....
~ ....
~ ....
+ ....
~ ....
log y(Hz) Fig. 2. Real part o f the c o m p l e x conductivity o-' versus f r e q u e n c y for L A S glass at 24.0°C. Points are experimental data. Lines are calculated f r o m the H N a n d K W W fit parameters in Eqs. (l 1) a n d (12).
fits predict qualitatively the observed fractional exponential dependence of ~r' on frequency in the high frequency range (oJ(~') >> 1): lim (dlog ~r'/dlog w) = 1 - (1 - c~)y for HN, (13a) lim (dlog o-'/dlog w) = 1 - 13 for KWW.
o) .-.-)oc
(13b) However, at low frequencies the HN fit, unlike the KWW fit, fails to predict the levelling off of the o-' versus f plot at the dc conductivity o'. Rather, at low frequencies the HN fit leads to a continual decrease of o-' with decreasing frequency with limiting slope: lim (dlog o " / d l o g w) = a.
(14)
co--~ 0
In similar fashion, the HN fit predicts at low frequencies a continual rise with decreasing frequency of the real part of the complex permittivity e', lim (dlog e ' / d l o g w) = c~- 1, ~o---*0
(15)
rather than a levelling off of e' at a static low frequency value, as predicted by the KWW fit and observed experimentally [1]. Finally, the complex plane resistivity plot of p" versus p' predicted from the HN fit does not converge on the dc resistivity p ( = 1/o-) at low frequencies, as observed experimentally and predicted by the KWW fit, but rather diverges with decreasing frequency. Because of the failure of the HN relaxation function to account correctly in a qualitative fashion for the observed low frequency electrical relaxation behavior, it must be judged pathological and unsuited for the description of electric field relaxation in ionically conducting materials.
3. Conclusions In agreement with the conclusions reached in our earlier paper [1], it appears that the KWW distribution of relaxation times remains the most accurate description of electrical relaxation in ionically con-
C.T. Moynihan / Journal of Non-Crystalline Solids 203 (1996) 359-363
ducting melts, glasses and crystals, if one wishes to describe the data in the frequency region where most of the relaxation is taking place in terms of a minimal number of parameters. We should emphasize, as before, that the value of the KWW/3 parameter is a description and not an explanation of the electrical relaxation process. The utility of the K W W / 3 parameter is that it can be used as a simple index for charting correlations and systematizing trends in the shapes of electrical relaxation curves as a function of variables such as temperature and glass or melt composition.
363
References [1] C.T. Moynihan, J. Non-Cryst. Solids 172-174 (1994) 1395. [2] P.B. Macedo, C.T. Moynihan and R. Bose, Phys. Chem. Glasses 13 (1972) 171. [3] C.T. Moynihan, L.P. Boesch and N.L. Laberge, Phys. Chem. Glasses 14 (1973) 122. [4] A. Bums, G.D. Chryssikos, E. Tombari, R.H. Cole and W.M. Risen Jr., Phys. Chem. Glasses 30 (1989) 264. [5] S. Havriliak and S. Negami, J. Polym. Sci. C 14 (1966) 99.